# Maximum Exponent of Boolean Circulant Matrices with Constant Number of Nonzero Entries in their Generating Vector

### Abstract

It is well-known that the maximum exponent that an $n$-by-$n$ boolean primitive circulant matrix can attain is $n-1$. In this paper, we find the maximum exponent attained by $n$-by-$n$ boolean primitive circulant matrices with constant number of nonzero entries in their generating vector. We also give matrices attaining such exponents. Solving this problem we also solve two equivalent problems: 1) find the maximum exponent attained by primitive Cayley digraphs on a cyclic group whose vertices have constant outdegree; 2) determine the maximum order of a basis for ${\Bbb Z}_{n}$ with fixed cardinality.